\documentclass[12pt,reqno]{article} \usepackage[usenames]{color} \usepackage{amssymb} \usepackage{graphicx} \usepackage{amscd} \usepackage{epsfig} \usepackage[colorlinks=true, linkcolor=webgreen, filecolor=webbrown, citecolor=webgreen]{hyperref} \definecolor{webgreen}{rgb}{0,.5,0} \definecolor{webbrown}{rgb}{.6,0,0} \usepackage{color} \usepackage{fullpage} \usepackage{float} \usepackage{graphics,amsmath,amssymb} \usepackage{amsthm} \usepackage{amsfonts} \usepackage{latexsym} \usepackage{epsf} \setlength{\textwidth}{6.5in} \setlength{\oddsidemargin}{.1in} \setlength{\evensidemargin}{.1in} \setlength{\topmargin}{-.5in} \setlength{\textheight}{8.9in} \newcommand{\seqnum}[1]{\href{http://www.research.att.com/cgi-bin/access.cgi/as/~njas/sequences/eisA.cgi?Anum=#1}{\underline{#1}}} \newcommand{\Aut}{\mbox{Aut}} \newcommand{\Out}{\mbox{Out}} \newcommand{\Ore}{\mbox{Ore}} \newcommand{\ord}{\mbox{ord}} \newcommand{\ind}{\mbox{ind}} \newcommand{\Res}{\mbox{Res}} \newcommand{\Gal}{\mbox{Gal}} \newcommand{\Eis}{\mbox{Eis}} \newcommand{\Fields}{\mbox{\rm Fields}} \newcommand{\TRFields}{\mbox{TRFields}} \newcommand{\Algebras}{\mbox{\rm Algebras}} \newcommand{\WildParts}{\mbox{\rm WildParts}} \newcommand{\WildPartitions}{\mbox{\rm WildPartitions}} \newcommand{\F}{\mathbb F} \newcommand{\cA}{\mathcal A} \newcommand{\cE}{\mathcal E} \newcommand{\cF}{\mathcal F} \newcommand{\cO}{\mathcal O} \newcommand{\cS}{\mathcal S} \newcommand{\cK}{\mathcal K} \newcommand{\cP}{\mathcal P} \newcommand{\cU}{\mathcal U} \newcommand{\Z}{\mathbb Z} \newcommand{\C}{\mathbb C} \newcommand{\Q}{\mathbb Q} \newcommand{\R}{\mathbb R} \begin{document} \begin{center} \epsfxsize=4in \leavevmode\epsffile{logo129.eps} \end{center} \begin{center} \vskip 1cm{\LARGE\bf Wild Partitions and Number Theory} \\ \vskip 0.5 cm \large David P. Roberts\\ Division of Science and Mathematics\\ University of Minnesota, Morris\\ Morris, MN, 56267 \\ USA\\ \href{mailto:roberts@morris.umn.edu}{\tt roberts@morris.umn.edu}\\ \end{center} \vskip .2in \begin{abstract} We introduce the notion of wild partition to describe in combinatorial language an important situation in the theory of $p$-adic fields. For $Q$ a power of $p$, we get a sequence of numbers $\lambda_{Q,n}$ counting the number of certain wild partitions of $n$. We give an explicit formula for the corresponding generating function $\Lambda_Q(x) = \sum \lambda_{Q,n} x^n$ and use it to show that $\lambda^{1/n}_{Q,n}$ tends to $Q^{1/(p-1)}$. We apply this asymptotic result to support a finiteness conjecture about number fields. Our finiteness conjecture contrasts sharply with known results for function fields, and our arguments explain this contrast. \end{abstract} \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}{Proposition}[section] \newtheorem{conjecture}{Conjecture}[section] \newtheorem{corollary}{Corollary}[section] \newtheorem{lemma}{Lemma}[section] \newtheorem{definition}{Definition}[section] \section{Introduction} \label{Intro} The sequence \seqnum{A000041} of integers $\lambda_n$ giving the number of partitions of $n$ is important throughout mathematics. Its generating function is \begin{equation} \label{part} \Lambda(x) = \sum_{n=0}^\infty \lambda_n x^n = \prod_{e=1}^\infty \frac{1}{1-x^e} = 1 + x + 2 x^2 + 3 x^3 + 5 x^4 + 7 x^5 + \cdots . \end{equation} In this paper, we consider for any prime power $Q = p^{n_0}$, an analogous integer sequence $\lambda_{Q,n}$ arising in a fundamental way in the number theory of $p$-adic fields. We evaluate the associated generating functions $\Lambda_Q(x) = \sum \lambda_{Q,n} x^n$, obtain corresponding asymptotics, and apply our results to support a finiteness conjecture about number fields. Generally speaking, our goal is to describe how a combinatorial viewpoint clarifies an important number-theoretic situation. The following two displays, incorporated in the sequence database \cite{Sl} as \seqnum{A131139} and \seqnum{A131140}, give a first sense of the functions $\Lambda_{Q}(x)$ and the corresponding sequences $\lambda_{Q,n}$: \begin{eqnarray*} \Lambda_2(x) & = & \frac{1}{1-x} \cdot \frac{1-x^2}{\left(1-2 x^2\right)^2} \cdot \frac{1}{1-x^3} \cdot \frac{ \left(1-x^4\right) \left(1-4 x^4\right)^2 }{\left(1-8 x^4\right)^4} \cdot \frac{1}{1-x^5} \cdot \frac{1-x^6}{\left(1-8 x^6\right)^2} \cdot \; \cdots \\ & = & 1 + x + 4 x^2 + 5 x^3 + 36 x^4 + 40 x^5 + 145 x^6 + 180 x^7 + 1572 x^8 + 1712 x^9 + \cdots ,\\ &\\ \Lambda_3(x) & = & \frac{1}{1-x} \cdot \frac{1}{1-x^2} \cdot \frac{\left(1-x^3\right)^2}{\left(1-3 x^3\right)^3} \cdot \frac{1}{1-x^4} \cdot \frac{1}{1-x^5} \cdot \frac{\left(1-x^6\right)^2}{\left(1-9 x^6\right)^3} \cdot \; \cdots \\ & = & 1 + x + 2 x^2 + 9 x^3 + 11 x^4 + 19 x^5 + 83 x^6 + 99 x^7 + 172 x^8 + 1100 x^9 + \cdots . \end{eqnarray*} In general, $\Lambda_Q(x)$, like its model $\Lambda(x)$, is given by a product over positive integers $e$. For $p$ not dividing $e$, the corresponding factor is $1/(1-x^e)$ again. However for $p$ dividing $e$, this factor is more complicated. In the number-theoretic context, the former factors reflect {\em tame} ramification and the latter reflect {\em wild} ramification. The paper is organized so its starts in combinatorics and ends in number theory. The main combinatorial objects, wild partitions, are defined so that they correspond bijectively to the main number theoretic objects, geometric classes of $p$-adic algebras. We do not pursue it here, but a future goal is to specify one bijection as the conventional one, so that the very simple objects, wild partitions, index the more complicated objects, geometric classes of $p$-adic algebras. Such a labeling of $p$-adic algebras could be incorporated into the database of local fields \cite{JR} and would considerably facilitate the $p$-adic analysis of number fields. Sections~\ref{Wild}-\ref{One} are our combinatoric sections. For a factorization $n_0 = e_0f_0$, we define {\em $(p,e_0,f_0)$-wild partitions} and a corresponding complicated three-variable generating function $\Lambda_{p,e_0,f_0}(x,y,z)$. Our definitions are not particularly motivated from a purely combinatoric point of view. Rather, as indicated above, they are chosen to mimic the structure of $p$-adic fields. For the sake of comparison, we consider first the specialization $(y,z) = (1,p^{-f_0})$ and get the remarkable simplification \begin{align} \label{serrespec} \Lambda_{p,e_0,f_0}(x,1,p^{-f_0}) & = \Lambda(x). \end{align} As we'll indicate, this identity is related to the Serre mass formula \cite{Se}. Our main interest is in the new specialization $(y,z) = (1,1)$. We define $\Lambda_Q(x)$ by an explicit formula and find \begin{align} \label{myspec} \Lambda_{p,e_0,f_0}(x,1,1) & = \Lambda_Q(x), \end{align} independently of the factorization $n_0 = e_0 f_0$. Our explicit formula allows us to consider arbitrary real powers $Q = p^\nu \geq 1$ so that $Q$ no longer determines $p$ and we accordingly write $\Lambda_{p,Q}(x)$. One has $\Lambda_{p,1}(x) = \Lambda(x)$, independently of $p$. Thus another point of view is that for each prime $p$ we have a $Q$-analog of $\Lambda(x)$. Sections~\ref{Core} and \ref{Asymptotics} consider analytic number theory associated to $\Lambda_{p,Q}(x)$. We express $\Lambda_{p,Q}(x)$ directly in terms of $\Lambda(x)$ and observe that as a consequence \begin{equation} \label{rootlimit} \lim_{n \rightarrow \infty} \lambda_{p,Q,n}^{1/n} = Q^{1/(p-1)}. \end{equation} Thus $\lambda_{p,Q,n}$ grows exponentially with growth factor $Q^{1/(p-1)}$. Equation~\eqref{rootlimit} contrasts with the famous Hardy-Ramanujan statement \cite{HR} of subexponential growth, $\lambda_{n} \sim e^{\pi \sqrt{2 n/3}}/(4 n \sqrt{3})$. It quantifies the extent to which wild ramification predominates over tame ramification in number theory. Another contrast between ordinary partitions and our $Q$-analogs is that $\lambda_n/\lambda_{n-1}$ tends to $1$ while $\lambda_{p,Q,n}/\lambda_{p,Q,n-1}$ has oscillatory behavior which becomes more pronounced as $Q$ increases. We conjecture an asymptotic of the form \begin{equation} \label{asymphope1} \lambda_{p,Q,n} \sim c_{p,Q}(n) C_p(Q) n^{B_p(Q)} e^{A_p(Q) \sqrt{n}} Q^{n/(p-1)}, \end{equation} with an explicit factor $c_{p,Q}(n)$ capturing the oscillatory behavior of $\lambda_{p,Q,n}/\lambda_{p,Q,n-1}$. Sections~\ref{RC}-\ref{adic2} are set in the framework of local algebraic number theory. The material here is somewhat more technical, but we have arranged our presentation so that the only prerequisite is familiarity with basic facts about $p$-adic fields. Section~\ref{RC} sets up the general situation and illustrates it for the fields $\R$ and $\C$, getting simple functions $\Lambda_\R(x)= e^x$ and $\Lambda_\C(x) = e^{x+x^2/2}$ which serve as analogs of our $\Lambda_Q(x)$. For Sections~\ref{Eisenstein}-\ref{adic2}, we let $F$ be an extension field of the $p$-adic field $\Q_p$, of ramification index $e_0$, inertial degree $f_0$, and thus degree $n_0 = e_0 f_0$ and residual cardinality $q = p^{f_0}$. Section~\ref{adic1} explains how the coefficient $\lambda_{n,c_t,c_w}$ of $x^n y^{c_t} z^{c_w}$ gives the ``total mass'' of algebras $K$ over $F$ having relative degree $n$, tame conductor $c_t$, and wild conductor $c_w$. Section~\ref{adic2} works over the maximal unramified extension $F^{\rm un}$ of $F$. It explains how $\lambda_{n,c_t,c_w}$ also counts extension algebras of $F^{\rm un}$ with the corresponding invariants. The perspective of Section~\ref{adic1} is more directly connected with the literature, while the perspective of Section~\ref{adic2} explains why the coefficients of $\Lambda_{p,e_0,f_0}(x,y,z)$ are integers. Summing over the possible $c_t$ and $c_w$, one gets that the total mass $\lambda_{F,n}$ of degree $n$ extension algebras of $F$ is $\lambda_{Q,n}$, where $Q = p^{n_0}$. Section~\ref{NumberFields} shifts to global algebraic number theory, working over an arbitrary number field $F$. It addresses a question raised in \cite{MR} on the size of sets $\mbox{Fields}^{\rm big}_{F,n,S}$ of relative degree $n$ number fields $K/F$. To be in $\mbox{Fields}^{\rm big}_{F,n,S}$, the extension $K/F$ must have associated Galois group $A_n$ or $S_n$ and ramification contained within the prescribed finite set of places $S$ of $F$ including the Archimedean places. A recent heuristic of Bhargava \cite{Bh} yields \begin{equation} \label{Bhargintro} \frac{1}{2} \prod_{v \in S} \lambda_{F_v,n} \end{equation} as a first guess (after slight modifications in degrees $\leq 3$) for the size of $\mbox{Fields}^{\rm big}_{F,n,S}$. The Archimedean factors $\lambda_{\R,n}$, $\lambda_{\C,n}$ decay superexponentially and we have proved that the remaining $\lambda_{F_v,n}$ grow only exponentially. Thus \eqref{Bhargintro} leads to the prediction that for fixed $(F,S)$, the set $\mbox{Fields}^{\rm big}_{F,n,S}$ is empty for sufficiently large $n$. In other words $\mbox{Fields}^{\rm big}_{F,S} = \coprod_n \mbox{Fields}^{\rm big}_{F,n,S}$ is finite. This finiteness statement has a certain irony to it: normally one considers $A_n$ and especially $S_n$ to be the ``generic expectation'' for Galois groups of number fields; the statement says that in the setting of prescribed ramification, these groups are in fact the exceptions. Finally, Section~\ref{Contrast} considers positive characteristic analogs of all the previous considerations. In positive characteristic, our finiteness statement fails very badly. The theory we present here explains this failure as due to two sources, either one of which suffices to void our argument for the finiteness statement. One source is that $Q$ must be considered $\infty$ and this forces all the $\lambda_{F_v,n}$ appearing in \eqref{Bhargintro} to be infinite for $n \geq p$. Another source is that there are no Archimedean places of $F$, and thus no superexponentially decaying factors in \eqref{Bhargintro}. Readers who want to quickly see the main ideas in a streamlined setting are invited to first focus on the special case $e_0 = f_0 = 1$. Then $n_0 = 1$ too, $p=q=Q$, and later $E = e$. The ground fields $F$ of Sections~\ref{RC}-\ref{adic2} are then only the completions of $\Q$, i.e.\ $\R$ and the $\Q_p$. The ground fields $F$ of Section~\ref{NumberFields} are then limited to simply $\Q$ itself. However by explicit examples with $n_0 = 2$ in Sections~\ref{Wild}, \ref{Three}, \ref{Asymptotics}, and \ref{NumberFields}, we try to assist readers in appreciating the case of general $(e_0,f_0)$. Sections~\ref{RC}-\ref{NumberFields} make clear that general $(e_0,f_0)$ is the natural setting from a number-theoretic point of view. The naturality of this setting is emphasized by Sections~\ref{Core} and \ref{Asymptotics} which interpolate $n_0 = e_0 f_0 \in \Z_{\geq 1}$ with general reals $\nu \geq 0$. The naturality of the general setting is further underscored by Section~\ref{Contrast}, which is based on the limiting case $e_0 = \infty$. \section{Wild partitions} \label{Wild} \paragraph{Basic notation.} An ordinary partition is an element of the free abelian monoid generated by the set of allowed parts $P = \{1,2,3,\dots\}$, for example \begin{equation} \label{exampleordinary} \mu_{\rm ordinary} = 9 + 7 + 3 + 3 + 2 + 2 + 2 + 1. \end{equation} Wild partitions are more complicated in two ways. First, the set $P$ is replaced by a set $P(p,e_0)$ mapping surjectively to $P$, with infinite fibers above multiples of $p$. Second, necessary for obtaining finiteness, an invariance condition with respect to an operator $\sigma = \sigma_p^{f_0}$ enters. As just indicated, our notion of wild partition depends not only on a prime $p$, but also on two positive integers $e_0$ and $f_0$. Let $q = p^{f_0}$ and $n_0 = e_0f_0$. Our notations $e_0$, $f_0$, $n_0$ and $q$ come from standard notations in the number-theoretic situation of Sections~\ref{Eisenstein}-\ref{adic2} inspiring our definitions. The quantity $Q = p^{n_0}$ is important for us, but does not have a standard number-theoretic notation. Strictly speaking, our sets of wild partitions depend also on a choice of algebraic closure $\overline{\F}_p$ of the prime field $\F_p = \Z/p \Z$. However all algebraic closures are isomorphic and so our final formulas counting certain wild partitions are independent of this choice. As usual, for a power $p^u$ of $p$ we denote by $\F_{p^u}$ the unique subfield of $\overline{\F}_p$ of cardinality $p^u$. We denote by $\sigma_{p}$ the Frobenius element $k \mapsto k^p$ in $\Gal(\overline{\F}_p/\F_p)$. Similarly, we denote by $\sigma_{p^u}$ the element $\sigma_p^{u}$; it is a topological generator of $\Gal(\overline{\F}_p/\F_{p^u})$. Most important for us is the operator $\sigma_q$, which we abbreviate by simply $\sigma$. We reserve $e$ for our main variable running over $P$. As a standing convention, we systematically write $e = p^w t$ with $p^w$ the largest power of $p$ dividing $e$. We think of $w$ as the {\em wildness} of $e$, $p^{w}$ as the {\em wild part} of $e$, and $t$ as the {\em tame part} of $e$. As another standing convention, we abbreviate $e_0 e$ by $E$. \paragraph{Ore numbers and their associated dimensions and spaces.} An important notion in number theory is the set of {\em Ore numbers} \begin{equation} \Ore(p,e_0,e) \subseteq \{0,1,\dots,w E - 1,w E\}. \end{equation} To understand the set $\Ore(p,e_0,e)$, it is convenient present it as an array, as for the case $(p,e_0,e) = (3,2,9)$ for which $w = 2$: \begin{equation} \label{orefirst} \begin{array}{ccccccccc} . & & & & & & & & \\ \hline . & 8 & 7 & . & 5 & 4 & . & 2 & 1 \\ . & 17 & 16 & . & 14 & 13 & . & 11 & 10 \\ \hline . & 26 & 25 & 24 & 23 & 22 & 21 & 20 & 19 \\ 36 & 35 & 34 & 33 & 32 & 31 & 30 & 29 & 28 \\ \end{array} \end{equation} In general, the array $\Ore(p,e_0,e)$ consists of a degenerate zeroth block, followed by $w$ full blocks. The zeroth block has only a single spot, filled by $0$ if $w=0$ and empty otherwise. The full blocks each have $e_0$ rows and $e$ columns. For $1 \leq j \leq w-1$, the $j^{\rm th}$ block consists of the integers in the interval $[(j-1) E+1,j E]$ which are not multiples of $p^j$. The $w^{\rm th}$ block consists of these entries together with $wE$. Considering the table as a whole, we refer to all the entries as {\em non-maximal}, except for $wE$ which is {\em maximal}. Our array format, including the right-to-left order, is intended to facilitate the discussion in Section~\ref{Eisenstein}, where the number-theoretic origin of $\Ore(p,e_0,e)$ is explained. An important quantity in our situation is the {\em dimension} $d(p,e_0,e,s)$ associated to an Ore number $s \in \Ore(p,e_0,e)$. It is the number of integers in $[0,s-1]$ which are not in $\Ore(p,e_0,e)$. Thus $d(3,2,9,20) = 7$, as there are seven omitted numbers less than $20$ on the displayed Ore table \eqref{orefirst}. The way dimensions arise in number theory is explained in Section~\ref{adic1}. An Ore number $s \in \Ore(p,e_0,e)$ determines a subset $W(p,e_0,e,s)$ of the vector space $\overline{\F}_p^{d(p,e_0,e,s)}$ as follows. For non-maximal $s$, the set $W(p,e_0,e,s)$ consists of the subset of vectors with non-zero first coordinate. In the maximal case $s = w E$, the subset $W(p,e_0,e,s)$ is defined to be all of $\overline{\F}_p^{d(p,e_0,e,s)}$. The Frobenius element $\sigma = \sigma_p^{f_0}$ acts coordinate-wise on each $W(p,e_0,e,s)$, as indeed $\sigma_p$ itself acts. The number of fixed points of $\sigma$ is clearly $q^{d(p,e_0,e,s)}(1-1/q)$ for non-maximal $s$ and $q^{d(p,e_0,e,s)}$ for the maximal $s = w E$. The explicit $\sigma$-sets $W(p,e_0,e,s)$ just introduced are isomorphic to less explicit $\sigma$-sets arising naturally in number theory, as explained in Section~\ref{adic2}. We use $s$ as our variable running over Ore numbers, because Ore numbers are also called Swan conductors. \paragraph*{Wild partitions and associated invariants.} We are now in a position to make the main definition of the combinatorial part of this paper. \begin{definition} \label{combdef} A $(p,e_0,f_0)$-wild partition is an element of the free abelian monoid on the set \begin{equation} P(p,e_0) = \coprod_{e \in \Z_{\geq 0}} \coprod_{s \in {\rm Ore}(p,e_0,e)} W(p,e_0,e,s) \end{equation} which is fixed by $\sigma = \sigma_p^{f_0}$. \end{definition} \noindent Usually $(p,e_0,f_0)$ is fixed and clear from the context. Then we just say ``wild partition'' rather than $(p,e_0,f_0)$-wild partition. We denote elements of $P(p,e_0)$ as doubly-subscripted integers $e_{s;\omega}$, with $s \in \Ore(p,e_0,e)$ and $\omega \in W(p,e_0,e,s)$. If $p$ does not divide $e$, then the only possible subscript is ``0;0'' and so we allow ourselves to omit it. As an example of our notation, let $i$ be one of the two square roots of $-1$ in $\F_9$. Then \begin{equation} \label{examplewild} \mu_{\rm wild} = 9_{20; 1,1,0,2,2,0,1} + 7 + 3_{1;i} + 3_{1;-i} + 2 + 2 + 2 + 1 \end{equation} is a wild partition for $(p,e_0,f_0) = (3,2,1)$. To check that $\mu_{\rm wild}$ is indeed formed according to our rules, note that $d(3,2,9,20) = |\{0,3,6,9,12,15,18\}| = 7$ from \eqref{orefirst}, and so it is proper that first subscripted $\omega$ has length $7$. Also the first coordinate of this $\omega$ is non-zero, as required. The Ore table for $(p,e_0,e) = (3,2,3)$ omits $0$ and has first row ``$\cdot \; 2 \; 1$'', so that $d(3,2,3,1) = |\{0\}| = 1$; thus $3_{1;i}$ and $3_{1;-i}$ are properly constructed wild parts. Finally $q = p^{f_0} = 3^1 = 3$ and so $\sigma(i) = i^3 = -i$; thus $\mu_{\rm wild}$ satisfies the $\sigma$-invariance condition. By definition, one can add wild partitions just as one can add ordinary partitions. Wild partitions have three obvious additive integer invariants, all important in the underlying number-theoretic situation. First, as for ordinary partitions, one has the {\em degree} $n$, defined as usual as the sum of the parts $e_i$. Second, defined but often not important for ordinary partitions, one has the {\em tame conductor $c_t$}, the sum of the $e_i-1$. Third, particular to our wild situation, one has the {\em wild conductor $c_w$}, the sum of the first subscripts $s_i$. Thus for the wild partition \eqref{examplewild}, one has $(n,c_t,c_w) = (29,21,22)$. \section{The generating functions $\Phi_{\cF}(x,y,z)$ and $\Lambda_{\cF}(x,y,z)$} \label{Three} Fix for this section a triple $(p,e_0,f_0)$ as in the previous section. These three quantities figure rather passively into our current considerations. We will have other more active quantities as well. Accordingly, we abbreviate via $\cF = (p,e_0,f_0)$. When we are continuing the example started in \eqref{exampleordinary}, \eqref{examplewild}, we will take $\cF = (3,2,1)$. \paragraph*{Irreducible and isotypical partitions.} We say a wild partition is {\em irreducible} if it is non-zero and cannot be written as the sum of two non-zero wild partitions. Every wild partition is uniquely the sum of its irreducible constituents. For example, $\mu_{\rm wild}$ has seven irreducible constituents, \begin{equation} \label{examplewild2} \mu_{\rm wild} = 9_{20; 1,1,0,2,2,0,1} + 7 + (3_{1;i} + 3_{1;-i})+ 2 + 2 + 2 + 1. \end{equation} We similarly say that a wild partition is {\em isotypical} if it has the form $m \mu$ for $\mu$ an irreducible wild partition and $m$ a positive integer. Every wild partition is uniquely the sum of its isotypical constituents. For example, $\mu_{\rm wild}$ has five isotypical constituents, \begin{equation} \label{examplewild3} \mu_{\rm wild} = 9_{20; 1,1,0,2,2,0,1} + 7 + (3_{1;i} + 3_{1;-i})+ (2 + 2 + 2)+ 1. \end{equation} If the wild partition $\mu$ is irreducible, we say that the isotypical wild partition $m \mu$ has mass $1/m$. In the number-theoretic settings of Sections~\ref{Eisenstein}-\ref{adic2}, wild partitions correspond to geometric packets of algebras of total mass one. A packet contains fields if and only if the corresponding wild partition is isotypical; in this case, the fields in the packets have total mass $1/m$ and the non-fields total mass $1-1/m$. \paragraph*{Definition of $\Phi_\cF(x,y,z)$ and $\Lambda_{\cF}(x,y,z)$.} Let $\phi_{\cF,n,c_t,c_w}$ be the { total mass} of $\cF$-wild isotypical partitions of degree $n$, tame conductor $c_t$, and wild conductor $c_w$. Then the corresponding generating function is expressible as a sum over irreducibles, \begin{eqnarray} \label{phi1} \Phi_{\cF}(x,y,z) & = & \sum_{n=0}^\infty \sum_{c_t=0}^\infty \sum_{c_w=0}^\infty \phi_{\cF,n,c_t,c_w} x^n y^{c_t} z^{c_w} \\ \label{phi2}& = & \sum_{\mu} \sum_{m=1}^\infty \frac{1}{m} x^{m n(\mu)} y^{m c_t(\mu)} z^{m c_w(\mu)} \\ & = & \label{phi3} \sum_{\mu} \log \left( \frac{1}{1 - x^{n(\mu)} y^{c_t(\mu)} z^{c_w(\mu)} } \right). \end{eqnarray} Similarly, let $\lambda_{\cF,n,c_t,c_w}$ be the { total number} of $\cF$-wild partitions of degree $n$, tame conductor $c_t$, and wild conductor $c_w$. Then its generating function is expressible as a product over irreducibles, \begin{eqnarray} \label{lam1} \Lambda_{\cF}(x,y,z) & = & \sum_{n=0}^\infty \sum_{c_t=0}^\infty \sum_{c_w=0}^\infty \lambda_{\cF,n,c_t,c_w} x^n y^{c_t} z^{c_w} \\ \label{lam2}& = & \prod_{\mu} \sum_{m=0}^\infty x^{m n(\mu)} y^{m c_t(\mu)} z^{m c_w(\mu)} \\ \label{lam3} & = & \prod_{\mu} \left( \frac{1}{1 - x^{n(\mu)} y^{c_t(\mu)} z^{c_w(\mu)} } \right). \end{eqnarray} The presence of $1/m$ in \eqref{phi2} and the absence of a corresponding factor in \eqref{lam2} reflects that \eqref{phi1}-\eqref{phi3} are in the setting of total mass while \eqref{lam1}-\eqref{lam3} are in the setting of total number. Comparison of \eqref{phi1}-\eqref{phi3} with \eqref{lam1}-\eqref{lam3} shows that one has an exponential formula \begin{equation} \label{PhiLambda} \Lambda_\cF(x,y,z) = \exp \left( \Phi_{\cF}(x,y,z) \right) . \end{equation} We will have analogous exponential formulas on the level of fields and algebras in the sequel. \paragraph*{Computation of $\Phi_\cF(x,y,z)$ and $\Lambda_{\cF}(x,y,z)$.} The degree $n$ of an isotypical partition factors into $ef$, where $e$ is the degree of any constituent part and $f$ is the number of such parts. Following the terminology of the number-theoretic situation from which we are abstracting, we call $e$ the {\em ramification index} and $f$ the {\em inertial degree}. In our continuing example, both $(3_{1,i}+3_{1,-i})$ and $(2+2+2)$ have degree $n=6$. The corresponding $(e,f)$ are $(3,2)$ in the first case and $(2,3)$ in the second. The tame conductor is always $c_t = (e-1) f$, thus $4$ in the first case and $3$ in the second. The Swan conductor $s$ of an isotypical partition is the first subscript on any of the parts, these first subscripts being all equal. One has $c_w = f s$, this equation being $2 = 2 \cdot 1$ in the first case and $0 = 3 \cdot 0$ in the second. Let $\phi_\cF(e,f,s)$ be the total mass of isotypical $\cF$-wild partitions with ramification index $e$, inertial degree $f$, and Swan conductor $s$. One has \begin{equation} \label{abstractKrasner} \phi_\cF(e,f,s) = \frac{1}{f} |W(p,e_0,e,s)^{\sigma^f}| = \frac{1}{f} q^{f \, d(p,e_0,e,s)} \left( 1 - \delta_{s}^{wE} q^{-f} \right). \end{equation} Here, as usual, $X^g$ is the set of fixed points of an operator $g$ on a set $X$. Also, to unify cases, $\delta_{s}^{wE}$ is $1$ in the non-maximal case $s 0} (1-Q^{(p^{j}-1)e/(p-1)} x^{p^{j} e}) \right)^{p^{j+1}-p^{j}} \left( \prod_{{\rm ord}_p(e) = 0} \frac{1}{1-Q^{(p^{j}-1)e/(p-1)} x^{p^{j} e}} \right)^{p^{j}} \\ \label{rightefactor} &=& \left( \prod_{{\rm ord}_p(e) > j} (1-Q^{(1-p^{-j})e/(p-1)} x^e) \right)^{p^{j+1}-p^{j}} \left( \prod_{{\rm ord}_p(e) = j} \frac{1}{1-Q^{(1-p^{-j})e/(p-1)} x^e} \right)^{p^{j}} \end{eqnarray} Let $e = p^w t$ as usual. Then the $e$-factor of the left side of \eqref{globalminus} is given by \eqref{wildformula} and the $e$-factor of the right side of \eqref{globalminus} is the product of \eqref{rightefactor} for $j = 0$, \dots, $w$. The $w$ numerator factors of \eqref{wildformula} match the numerator factors in \eqref{rightefactor} for $j = 0, \dots, w-1$. The denominator factor of \eqref{wildformula} matches the denominator factor in \eqref{rightefactor} for $j=w$. \qed \section{Asymptotics} \label{Asymptotics} In this section we study the asymptotic behavior of the coefficients $\lambda_{p,Q,n}$ of the power series $\Lambda_{p,Q}(x)$. \paragraph*{A change of variables.} Abbreviate $Q^{-1/(p-1)}$ by $r \in (0,1]$ and change variables via $x = ry$. Define $\underline{\Lambda}_{p,Q}(y) = \Lambda_{p,Q}(r y) = \sum_{n=0}^\infty \underline{\lambda}_{p,Q,n} y^n$ so that $\underline{\lambda}_{p,Q,n} = r^n \lambda_{p,Q,n}$. Then \eqref{globalminus} takes on the simpler form \begin{equation} \label{globalminus2} \underline{\Lambda}_{p,Q}(y) = \prod_{j=0}^\infty \Theta_p(r y^{p^{j}})^{p^{j}}. \end{equation} As the radius of convergence of $\Theta_p(x)$ is $1$, the radius of convergence of the $j^{\rm th}$ factor is $r^{-1/p^{j}}$. So as $j$ increases from $1$, these radii decrease monotonically from $1/r$ with limit $1$. \begin{figure}[tbh] \begin{center} \epsfig{file = cascades.eps,width=6in} %\includegraphics[width=6in]{cascades} \parbox{5in}{\caption{\label{cascades} {Points $(n,\log(\underline{\lambda}_{2,2,n; k}))$ on the left and $(n,\log( \underline{\lambda}_{3,3,n; k}))$ on the right, with $k$ indicated by text.}} } \end{center} \end{figure} We indicate with a subscript $k$ the corresponding objects where the product in \eqref{globalminus2} is taken from $0$ to $k$. For any given $k$, the coefficients $\underline{\lambda}_{p,Q,n;k}$ eventually decay exponentially, by the above radius of convergence remarks. Figure~\ref{cascades} illustrates the decay. \paragraph*{Radius of convergence and upper bounds for $\lambda_{p,Q,n}$.} We are interested more in the behavior of the coefficients $\underline{\lambda}_{p,Q,n}$ themselves, rather than the cutoff versions $\underline{\lambda}_{p,Q,n; k}$. In general, let $\Theta(z)$ be any power series convergent at least on the closed disk of radius $r$ with $\Theta(0)=1$ and $|\Theta(r)|>1$. Then it is elementary that $\prod_{j=0}^\infty \Theta(r y^{p^j})^{p^j}$ converges for $|y|<1$. The product does not converge for $y=1$ because it formally has the form $\Theta(r)^\infty$. Thus our $\underline{\Lambda}_{p,Q}(y)$ have radius of convergence exactly one. In terms of Figure~\ref{cascades}, the upper envelope of the points plotted grows at most sub-linearly. In terms of the original $\lambda_{p,Q,n}$, one eventually has $\lambda_{p,Q,n} < Q^{n/(p-1)+\epsilon}$ for any positive $\epsilon$. \paragraph*{Root growth.} Sharpening the statement $\limsup \lambda_{p,Q,n}^{1/n} = Q^{1/(p-1)}$ just observed is the following. \begin{proposition} \label{rootprop} ${\displaystyle \lim_{n \rightarrow \infty} \lambda_{p,Q,n}^{1/n} = Q^{1/(p-1)}}$. \end{proposition} \proof We need only show that the sequence $\lambda_{p,Q,n}^{1/n}$ has no limit points smaller than $Q^{1/(p-1)}$. In other words, we need only show that the sequence $\underline{\lambda}_{p,Q,n}^{1/n}$ has no limit points smaller than $1$. From \eqref{pcore}, we know that the expansion of $\Theta_p(x)$ begins with $\sum_{n=0}^{p-1} \lambda_n x^n$ which is term-by-term at least $\sum_{n=0}^{p-1} x^n$. The $j^{\rm th}$ factor $\Theta_p(r y^p)^{p^j}$ of \eqref{globalminus} is coefficient-wise bounded from below by $\Theta_p(r y^p)$ which is in turn coefficient-wise bounded from below by $\sum_{a=0}^{p-1} r^a y^{a p^j}$. So $\Lambda_{p,Q}(y)$ is coefficient-wise bounded from below by $\prod_{j=0}^\infty \sum_{a=0}^{p-1} r^a y^{a p^j}$. If $n = \sum_{i=0}^{\log_p(n)} a_i p^i$ with $0 \leq a_i \leq p-1$, then \begin{equation*} \underline{\lambda}_{p,Q,n} \geq \prod_{i=0}^{\log_p(n)} r^{a_i} \geq r^{(1 + \log_p(n))(p-1)} \end{equation*} so that \begin{equation} \label{rootlowerbound} \underline{\lambda}^{1/n}_{p,Q,n} \geq r^{(1 + \log_p(n))(p-1)/n}. \end{equation} The right side of \eqref{rootlowerbound} tends to $1$ with $n$, proving that indeed $\underline{\lambda}_{p,Q,n}^{1/n}$ has no limit points smaller than $1$. \qed \paragraph*{Expected refined asymptotics.} Proposition~\ref{rootprop} is more than we need to support Conjecture~\ref{mainconj} below. However the extreme crudeness of the bounds in its proof suggests that stronger statements are provable. Rather than proceed incrementally, in the rest of this section we present numerical evidence and heuristic argument leading up to the very strong statement \eqref{asymphope2}, conjecturally extending the Hardy-Ramanujan asymptotic for $\lambda_{n}$. Another model of the type of statement sought is given in the next section for analogous quantities $\lambda_{F,n}$ for $F = \R$ and $F = \C$. There, \eqref{ArchAsymp2} gives an asymptotic equivalent to the root decay factor $\lambda_{F,n}^{1/n}$, analogous to our Proposition~\ref{rootprop}. More subtly, \eqref{ArchAsymp2} also says that the ratio decay factor $\lambda_{F,n}/\lambda_{F,n-1}$ has the same asymptotic equivalent. Finally \eqref{ArchAsymp1} is the sharpest statement, giving asymptotic equivalents to $\lambda_{F,n}$ itself. \paragraph*{Ratio oscillation.} In contrast to both ordinary partitions and the Archimedean cases, evidence strongly suggests that $\lambda_{p,Q,n}/\lambda_{p,Q,n-1}$ does not have a limiting value for $Q > 1$. Figure~\ref{startgrowth} graphs $\log(\underline{{\lambda}}_{p,Q,n})$ for \begin{figure}[tbh] \begin{center} \epsfig{file = twostarts.eps,width=6in} %\includegraphics[width=6in]{twostarts} \parbox{5in}{\caption{\label{startgrowth} Points $(n,\log(\underline{\lambda}_{2,2,n}))$ on the left and $(n,\log(\underline{\lambda}_{3,3,n}))$ on the right, for $0 \leq n \leq 100$. Also points $(n,\log(\underline{\lambda}_{2,4,n}))$ on the left and $(n,\log(\underline{\lambda}_{3,9,n}))$ on the right, for $36 \leq n \leq 100$. The oscillatory behavior modulo eight on the left and modulo nine on the right matches Table~\ref{cvalues} well.}} \end{center} \end{figure} $p=2,3$ and $Q=p^j$ with $j=1,2$. One sees smooth growth with oscillatory behavior superimposed. The corresponding picture for $n$ out through $4000$ shows no damping. In general, oscillatory behavior is barely visible for $Q$ near $1$ and increases in amplitude with $Q$. There seem to be dominant oscillations with period $p$, secondary oscillations with period $p^2$, tertiary oscillations with period $p^3$, and so on. The situation clearly calls for a Fourier analysis. \begin{figure}[tbh] \begin{center} \epsfig{file=twolambdas.eps, width=6in} %\includegraphics[width=6in]{twolambdas} \parbox{5in}{\caption{\label{lambdacontours} Contour plots of $|\underline{\Lambda}_{2,2}(y)|$ on the left and $|\Lambda_{3,3}(y)|$ on the right, both on the open unit disk $|y|<1$, with dark shading indicating large values. }} \end{center} \end{figure} Figure~\ref{lambdacontours} illustrates with two examples the magnitude of $\underline{\Lambda}_{p,Q}(y)$ as a function of the complex variable $y$. The needed Fourier analysis is connected with the limiting behavior of $\underline{\Lambda}_{p,Q}(y)$ as $|y|$ increases to $1$. For $r<1$, consider the function \begin{equation} \label{chat} \hat{c}_{p,Q}(y) = \prod_{j=0}^\infty \left( \frac{\Theta_p(r y^{p^j} )}{\Theta_p(r)} \right)^{p^j} \end{equation} defined on the unit circle. One has $\hat{c}_{p,Q}(1) = 1$ as indeed all factors are $1$. For $y$ satisfying $y^{p^k}=1$ but not $y^{p^{k-1}}=1$ the infinite product reduces to the product of its first $k$ factors, all of which are non-zero with absolute value less than one; thus $0 < |\hat{c}_{p,Q}(y)| < 1$ for these $y$. Finally if $y$ is otherwise, the infinite product converges to $0$. We view $\hat{c}_{p,Q}(y)$ as giving the normalized boundary values of $\underline{\Lambda}_{p,Q}(y)$. Intuitively, we can view $\underline{\Lambda}_{p,Q}(y)$ as having its most important singularity at $1$. This singularity is echoed in quantitatively smaller singularities at primitive $p^{\rm th}$ roots of unity. It is echoed in still smaller singularities at primitive roots of unity of order $p^2$, and so on, as illustrated by Figure~\ref{lambdacontours}. Table~\ref{boundvalues} gives a more numerical illustration, and includes also the cases $(p,Q) = (2,4)$ and $(p,Q) = (3,9)$. For given $p$, the echoes decay more slowly with larger $Q$. \begin{table}[tbh] \[ \begin{array}{|lll|c|lll|} \cline{1-3} \cline{5-7} \alpha & \hat{c}_{2,2}(e^{2 \pi i \alpha}) & \hat{c}_{2,4}(e^{2 \pi i \alpha}) & & \alpha & \hat{c}_{3,3}(e^{2 \pi i \alpha}) & \hat{c}_{3,9}(e^{2 \pi i \alpha}) \\ \cline{1-3} \cline{5-7} 0/1 & 1. &1. & \;\;\;\;\;\; & 0/1 &1. &1. \\ 1/8 & 0.0005 + 0.0009 i&0.0567 + 0.0405 i&& 1/9 & 0.0004 + 0.0010 i &0.0542 + 0 .0569 i\\ 1/4 & 0.0341 + 0.0130 i &0.2660 + 0.0624 i &&2/9&0.0006 + 0.0002 i&0.0560 - 0.0 029 i \\ 3/8 & 0.0007 - 0.0001 i&0.0490 - 0.0119 i &&1/3&0.1191 + 0.0210 i &0.4476 + 0.0 718 i\\ 1/2 & 0.2385 &0.5803 &&4/9&0.0007 + 0.0001 i&0.0463 + 0.020 5 i\\ \cline{1-3} \cline{5-7} \end{array} \] \begin{center} \parbox{5in}{\caption{\label{boundvalues} Some normalized boundary values $\hat{c}_{p,Q}(y)$ of $\underline{\Lambda}_{p,Q}(y)$, rounded to the nearest ten-thousandth. Only values in the upper half plane are given, because $\hat{c}_{p,Q}(\bar{y}) = \overline{\hat{c}_{p,Q}(y)}$.}} \end{center} \end{table} The function $\hat{c}_{p,Q}(y)$ enters into our Fourier analysis as follows. Taking an inverse Fourier transform, define \begin{equation} \label{inversetransform} c_{p,Q}(n) = \sum_{y} y^{-n} \hat{c}_{p,Q}(y), \end{equation} the sum being over all $p^{\rm th}$ power roots of unity. One can check that the sum in \eqref{inversetransform} indeed converges. Moreover, let $\Z_p$ be the $p$-adic integers, i.e.\ the completion of $\Z$ with respect to the sequence of finite quotients $\Z/p^j \Z$. Then $c_{p,Q}$, thought of as a function from $\Z$ to $\R$, extends continuously to a function from $\Z_p$ to $\R$. Table~\ref{cvalues} gives some values. \begin{table}[tbh] \[ \begin{array}{|rrr|c|rrr|} \cline{1-3} \cline{5-7} n & c_{2,2}(n) & c_{2,4}(n) & \;\;\;\;\;\;\;\;\; & n & c_{3,3}(n) & c_{3,9}(n) \\ \cline{1-3} \cline{5-7} 0 & 1.309 & 2.324 && 0 & 1.242 & 2.208 \\ 1 & 0.788 & 0.596 && 1 & 0.918 & 0.774 \\ 2 & 1.172 & 1.153 && 2 & 0.847 & 0.496 \\ 3 & 0.737 & 0.324 && 3 & 1.238 & 1.878 \\ 4 & 1.304 & 1.901 && 4 & 0.918 & 0.679 \\ 5 & 0.787 & 0.493 && 5 & 0.845 & 0.426 \\ 6 & 1.168 & 0.944 && 6 & 1.235 & 1.600 \\ 7 & 0.734 & 0.265 && 7 & 0.915 & 0.577 \\ \cline{1-3} \multicolumn{3}{c}{\;} & & 8 & 0.842 & 0.362 \\ \cline{5-7} \end{array} \] \begin{center} \parbox{5in}{\caption{\label{cvalues} Some values of $c_{p,Q}(n)$, rounded to the nearest thousandth. To the nearest thousandth, $c_{2,2}(n)$ depends only on $n$ modulo $8$ while $c_{3,3}(n)$ depends only on $n$ modulo $9$. Similarly, to the nearest hundredth, $c_{2,4}(n)$ and $c_{3,9}(n)$ depend only on $n$ to the respective moduli $8$ and $9$. }} \end{center} \end{table} We expect that the function $c_{p,Q}(n)$ fully captures the oscillatory behavior in the sense that \begin{equation} \label{ratiolimit} \frac{\underline{\lambda}_{p,Q,n}/c_{p,Q}(n)}{\underline{\lambda}_{p,Q,n-1}/c_{p,Q}(n-1)} \sim Q^{1/(p-1)}. \end{equation} Computations such as those illustrated by Figure~\ref{endgrowth} support this expectation. \paragraph*{Towards an asymptotic equivalent to $\lambda_{p,Q,n}$.} The smooth part of $\lambda_{p,Q,n}$ presents more of a mystery. Computations are consistent with the conjecture given as \eqref{asymphope1} in the introduction, namely \begin{equation} \label{asymphope2} \lambda_{p,Q,n} \sim c_{p,Q}(n) C_p(Q) n^{B_p(Q)} e^{A_p(Q) \sqrt{n}} Q^{n/(p-1)} \end{equation} for quantities $A_p(Q)$, $B_p(Q)$, and $C_p(Q)$ to be thought of as functions on the $Q$-interval $[1,\infty)$. For $Q = 1$, the oscillatory factor $c_{p,Q}(n)$ reduces to $1$. One has \[ (A_p(1), B_p(1), C_p(1)) = (\pi \sqrt{\frac{2}{3}} , -1, \frac{1}{4 \sqrt{3}}) \approx (2.56,-1.00,0.144), \] independently of $p$, by the Hardy-Ramanujan asymptotic for partitions \cite{HR}. Figure~\ref{endgrowth} graphs functions $A_p(Q) \sqrt{n} + B_p(Q) \log{n} + \log C_p(Q)$ with $(A_p(Q),B_p(Q),C_p(Q))$ deduced from a least squares fit to $\log(\underline{\lambda}_{p,Q,n}/c_{p,Q}(n))$ over $[20,4000]$. The drawn lines are thick enough so that they contain all the actual points $(n,\log(\underline{\lambda}_{p,Q,n}/c_{p,Q}(n)))$. For the most important quantity $A_p(Q)$, the fit yields \begin{align*} A_2(2) & \approx 1.66 \, , & A_3(3) & \approx 1.68 \, ,\\ A_2(4) & \approx 1.18 \, , & A_3(9) & \approx 1.21 \, . \end{align*} Numeric computations are not accurate enough to suggest an analytic form for $A_p(Q)$, $B_p(Q)$, and $C_p(Q)$; a more theoretical approach is needed. \begin{figure}[tbh] \begin{center} \epsfig{file=endgrowth.eps,width=6in} %\includegraphics[width=6in]{endgrowth3} \parbox{5in}{ \caption{\label{endgrowth} Least square fits to points $(n,\log(\underline{\lambda}_{2,2^j,n}/c_{2,2^j}(n)))$ on the left and $(n,\log(\underline{\lambda}_{3,3^j,n}/c_{3,3^j}(n)))$ on the right, for $j \in \{1,2\}$ and $20 \leq n \leq 4000$.} } \end{center} \end{figure} \section{Fields, algebras, and the cases $F = \R$ and $F = \C$} \label{RC} In this section, we introduce some concepts associated to field and algebra extensions of a given ground field $F$. These concepts will play a major role in the rest of the paper. Also we illustrate these concepts with the ground fields $F = \R$ and $F=\C$. These ground fields are particularly simple and familiar. Moreover, they play an essential role in the global considerations of Section~\ref{NumberFields}. \paragraph*{Fields and algebras.} For $F$ a field, let $\Fields_{F,n}$ be the set of isomorphism classes of separable degree $n$ field extensions of $F$. Our main interest is in characteristic zero, where all fields are separable; accordingly we drop the adjective ``separable.'' Similarly, let $\Algebras_{F,n}$ be the set of isomorphism classes of degree $n$ algebra extensions which are products of field extensions. For both fields and algebras, we allow ourselves also to drop the phrase ``of isomorphism classes'' since it always understood. Similarly we write just $K$ instead of say $[K]$ to indicate the isomorphism class of an algebra $K$. An algebra $K$ has an automorphism group $\Aut(K/F)$. We define, as is standard, its mass to be $1/|\Aut(K/F)|$. For some fields $F$, all the sets $\Fields_{F,n}$ are finite. Exactly in this case, all the larger sets $\Algebras_{F,n}$ are finite too. For these $F$, we define $\phi_{F,n}$ and $\lambda_{F,n}$ to be the total masses of $\Fields_{F,n}$ and $\Algebras_{F,n}$ respectively. Let \begin{align*} \Phi_{F}(x) & = \sum_{n=1}^\infty \phi_{F,n} x^n = x + \cdots, & \Lambda_{F}(x) & = \sum_{n=0}^\infty \lambda_{F,n} x^n = 1 + x + \cdots \end{align*} be the corresponding generating functions. Then one has the exponential formula \cite[Chapter 5]{St} \begin{equation} \label{expformula} \Lambda_{F}(x) = \exp(\Phi_{F}(x)), \end{equation} from the definition of mass and the way algebras are built from fields. \paragraph*{The cases $F=\R$ and $F=\C$.} With the above definitions, $\Fields_{\R,1} = \{\R\}$, $\Fields_{\R,2} = \{\C\}$, and otherwise $\Fields_{\R,n} = \emptyset$. Also $\Algebras_{\R,n} = \{\R^{r} \C^s : r + 2 s = n\}$, with the mass of $\R^r\C^s$ being $1/(r! s! 2^s)$. Even more simply, the only non-empty $\Fields_{\C,n}$ is $\Fields_{\C,1} = \{\C\}$. One has $\Algebras_{\C,n} =\{\C^n\}$, with the mass of $\C^n$ being $1/n!$. Thus \begin{align} \label{functR} \Lambda_\R(x) & = \sum_{n=0}^\infty \lambda_{\R,n} x^n && = e^{x+x^2/2} && = 1 + x + \frac{2}{2} x^2 + \frac{4}{6} x^3 + \frac{10}{24} x^4 + \frac{26}{120} x^5 +\cdots , \\ \label{functC} \Lambda_\C(x) & = \sum_{n=0}^\infty \lambda_{\C,n} x^n && = e^x && = 1 + x + \frac{1}{2} x^2 + \frac{1}{6} x^3 + \frac{1 }{24} x^4 + \frac{1}{120} x^5 + \cdots . \end{align} The numbers $n! \lambda_{\R,n}$ form the sequence \seqnum{A000085} giving, among other interpretations, the number of involutions in the symmetric group $S_n$. One has the asymptotic formulas \begin{align} \label{ArchAsymp1} \lambda_{\R,n} & \sim \frac{e^{\frac{n}{2}+\sqrt{n}-\frac{1}{4}} n^{-\frac{n}{2}-\frac{1}{2}}}{2 \sqrt{\pi }} , & \lambda_{\C,n} & \sim \frac{e^{n}}{ n^{n+\frac{1}{2}} \sqrt{2 \pi }}, \end{align} the first due to Moser and Wyman \cite{MW} and the second being Stirling's approximation. On the level of ratio and root behavior, one has \begin{align} \label{ArchAsymp2} \frac{\lambda_{\R,n}}{\lambda_{\R,n-1}} \sim \lambda_{\R,n}^{1/n} & \sim \sqrt{\frac{e}{n}}, & \frac{\lambda_{\C,n}}{\lambda_{\C,n-1}} \sim \lambda_{\C,n}^{1/n} & \sim \frac{e}{n}, \end{align} thus superexponential decay. \section{Eisenstein polynomials} \label{Eisenstein} For this and the next two sections, fix a prime number $p$. Let $\Q_p$ be the field of $p$-adic numbers. Its ring of integers $\Z_p$ already arose naturally in Section~\ref{Asymptotics}. The maximal ideal of $\Z_p$ is generated by the prime number $p$, and the corresponding residue field is $\F_p = \Z_p/p$. For background on $p$-adic numbers, see e.g.\ \cite{Go}. We need mainly the algebraic theory of finite degree field extensions of $\Q_p$, i.e.\ Chapter~5 of \cite{Go}. \paragraph*{The ground field $F$.} For this and the next two sections, fix also an extension field $F$ of degree $n_0$ over $\Q_p$. So $F$ can be presented as $\Q_p[x]/g_0(x)$ for some irreducible polynomial $g_0(x)$ in $\Q_p[x]$ of degree $n_0$. We have no need to consider $g_0(x)$ again, as we will simply regard $F$ as given. Let $\cO$ be the ring of integers of $F$, let $\Pi$ be its maximal ideal, and let $\kappa = \cO/\Pi$. The ramification index of $F/\Q_p$ is the positive integer $e_0$ such that $\Pi^{e_0} = (p)$. The inertial degree of $F/\Q_p$ is the positive integer $f_0$ such that $q := |\kappa| = p^{f_0}$. One has $e_0 f_0 = n_0$. It is often clearer to avoid the language of ideals. To do this we fix a uniformizer $\pi$ of $F$, i.e.\ a generator of $\Pi$. For $a \in \cO - \{0\}$, we write $\ord_{\pi}(a) = b$ to mean that $a$ generates the ideal $(\pi^b)$. We define $\ord_\pi$ on all of $\cO$ by writing $\ord_\pi(0) = \infty$. \paragraph*{Extensions of $F$ and their numerical invariants.} Likewise, one can consider field extensions $K = F[x]/g(x)$ of $F$. The degree $n$ of such an extension factors into its ramification index $e$ and its inertial degree $f$. Another important invariant of a field extension $K/F$ is its discriminant $d(K/F)$, which is an ideal $\Pi^c$ in $\cO$. We focus on the discriminant-exponent $c$, which we call the conductor. If $K = F[x]/g(x)$, then the ideal generated by the polynomial discriminant \begin{equation} \label{discform} D(g) = (-1)^{n(n-1)/2} \Res_x(g(x),g'(x)) \end{equation} has the form $\Pi^{c + 2d}$ for $d$ a non-negative integer, called the defect of $g(x)$. The conductor of $K/F$ is naturally written as $c = c_t + c_w$, where $c_t$ is the tame conductor and $c_w$ is the wild conductor. Very simply, $c_t = f (e-1)$. The wild conductor $c_w$ is more complicated, but has the form $f s$, where $s \in \Ore(p,e_0,e)$ is a non-negative integer called the Swan conductor, as detailed below. We seek to understand the sets $\Fields_{F,n}$ introduced in the previous section. The decomposition \begin{equation} \Fields_{F,n} = \coprod_{ef=n} \coprod_{s \in {\rm Ore}(p,e_0,e)} \Fields_{F}(e,f,s) \end{equation} is a natural starting point. In the rest of this section, we explain how Eisenstein polynomials give an explicit understanding of the totally ramified part $\Fields_{F}(e,1,s)$. The cases $f > 1$ are easily reduced to the case $f=1$, as explained in the next section. \paragraph*{Eisenstein polynomials.} Consider monic polynomials of degree $e$ with coefficients in $\cO$. Such a polynomial \begin{equation} \label{eispoly} g(x) = x^e + a_{e-1} x^{e-1} + \cdots + a_{1} x + a_0 \end{equation} is called an Eisenstein polynomial if and only if $\pi$ divides all the coefficients $a_i$ and moreover $\pi^2$ does not divide $a_0$. Let $\Eis(\cO,e)$ be the space of degree $e$ Eisenstein polynomials over $\cO$. If $g(x)$ is an Eisenstein polynomial then $K = F[x]/g(x)$ is a totally ramified field extension of $F$. Moreover $\cO[x]/g(x)$ is its ring of integers which means that the defect $d$ of $g(x)$ is zero. The element $x \in \cO[x]/g(x)$ is a uniformizer, meaning that it generates the maximal ideal of $\cO[x]/g(x)$. Conversely, suppose a totally ramified $K$ is given. Then one can consider for each of its uniformizers $\omega$ the characteristic polynomial $g_\omega(x)$ of $\omega$ acting by multiplication on $K$, where $K$ is considered as an $e$-dimensional vector space over $F$. The resulting map \begin{equation} \label{keycover} \omega \mapsto g_\omega(x) \end{equation} is $|\Aut(K/F)|$-to-$1$ over its image $\Eis(\cO,e)_K \subseteq \Eis(\cO,e)$. \paragraph*{Conductors of Eisenstein polynomials.} The conductor $c = c_t + c_w$ of the Eisenstein polynomial \eqref{eispoly} is $\ord_x(g'(x))$, where $x$ here is understood as the given uniformizer of $\cO[x]/g(x)$ \cite[III.6] {SeCL}. The tame conductor $c_t$ is $e-1$. Thus \begin{equation} \label{eiscond} c_w = \ord_x(g'(x)) - (e-1). \end{equation} For $i=1$, \dots, $e$, define the $i^{\rm th}$ index of an Eisenstein polynomial \eqref{eispoly} to be \[ \ind_i(g(x)) = \ord_x(i a_i x^{i-1}) - (e-1) = e \; \ord_\pi(i a_i) + i - e = \left\{ \begin{array}{ll} {\displaystyle e \; \ord_\pi(i \frac{a_i}{\pi}) + i ,}& \mbox{if $i 1$. For example, let $L$ be a product of $m$ factors of $F^{\rm un}$. Then the geometric packet of models for $L$ consists of algebras $F_\mu$, where $\mu$ is a partition of $m$. Here if $\mu = \mu_1+\dots+\mu_h$ then $F_\mu = F_{\mu_1} \times \cdots \times F_{\mu_h}$, where, as before, $F_f$ denotes the degree $f$ unramified extension of $F$. Then $|\Aut(F_\mu/F)| = \prod_k k^{m_k} m_k!$, where $m_k$ is the number of times $k$ appears in $\mu$ and Equation~\eqref{mass1} becomes the class equation for the symmetric group $S_m$. When a packet $\{K_1,\dots,K_g\}$ contains a totally ramified field then all its elements are totally ramified fields. The database of local fields \cite{JR} contains many instances. For example, suppose $K_1$ is a sextic field with automorphism group $S_3$ and hence mass $1/6$. Then its packet is $\{K_1,K_2,K_3\}$ where $K_2$ has mass $1/2$ and $K_3$ has mass $1/3$. For $i=1,2,3,$ the corresponding Galois closures $K_i^g$ have Galois group $\Gal(K_i^g/F)$ with $S_3$ as inertia subgroup and the cyclic group $C_i$ as corresponding quotient. For $F = \Q_3$, the database presents five such packets. The packets just discussed correspond to the irreducible partitions of Section~\ref{Wild}. More generally suppose a packet $\{K_1,\dots,K_g\}$ contains a field with residual degree $f$. Then the $K_i$ which are fields all have residual degree $f$ and their total mass is $1/f$. These packets correspond to the isotypical partitions of Section~\ref{Wild}. \section{Number fields} \label{NumberFields} \paragraph*{The sets $\mbox{\rm $\Fields$}_{F,n,S}$.} Let $F[x] = \Q[x]/g_0(x)$ be a number field. Let $\cS(F)$ be its set of places, indexing the set of completions of $F$. Thus $\cS(\Q) = \{\infty,2,3,5,\dots\}$ and $\cS(F)$ maps surjectively to $\cS(\Q)$. For $S \subseteq \cS(F)$, let $\Fields_{F,n,S}$ be the set of isomorphism classes of degree $n$ field extensions $K/F$ ramified entirely within $S$. The ramification condition is then that for all $v \in \cS(F)-S$, and all $w \in \cS(K)$ over $S$, the local extension $K_w/F_v$ is unramified. In this context, we view $\C/\R$ as ramified. An extension $K/F$ has a Galois closure $K^g$ and hence a Galois group $\Gal(K^g/F)$; if $K = F[x]/g(x)$, then $K^g$ is by definition a splitting field of $g(x)$. The largest that $\Gal(K^g/F)$ can be is the full symmetric group $S_n$, with $n = [K:F]$. For $n \geq 3$, the second largest that $\Gal(K^g/F)$ can be is the alternating group $A_n$. We have a decomposition \begin{equation} \label{globaldecomp} \Fields_{F,n,S} = \Fields^{\rm sym}_{F,n,S} \coprod \Fields^{\rm alt}_{F,n,S} \coprod \Fields^{\rm small}_{F,n,S} \end{equation} and also write $\Fields^{\rm big}_{F,n,S}$ to indicate the union of the first two parts. We write also $\Fields^s_{F,S} = \coprod_n \Fields^s_{F,n,S}$ for any superscript $s$. In practice, it is easy to decide whether a given field $F[x]/g(x)$ has big or small Galois group. One quick way is to factor $g(x)$ in sufficiently many completions $F_v$ and use information from the degrees of the factor fields $K_w$; for most $v$, this reduces to a calculation in the residue field of $F_v$. For $n \geq 8$, a group-theoretical result of Jordan \cite{Jo} suffices: the Galois group is big if and only if \begin{equation} \label{Jordan} \mbox{the degree of $K_w/F_v$ is a prime in $(n/2,n-2)$} \end{equation} for some $K_w/F_v$. Many other criteria can be brought to bear as well. The computations are guided by the principle that the factor partitions for $v$ not ramified in $K$ are equidistributed in the set of partitions of $n$ according to the measure induced from the Haar measure on $\Gal(K^g/F)$. We are interested in the case of $S$ finite. Then a classical fact is that the sets $\Fields_{F,n,S}$ are all finite. Analogously to the local situation, it is natural to define the mass of a field $K$ to be $1/|\Aut(K/F)|$. From \eqref{globaldecomp} we have $\phi_{F,n,S} = \phi^{\rm big}_{F,n,S} + \phi^{\rm small}_{F,n,S}$. Our main concern is $\phi^{\rm big}_{F,n,S}$, which is just the cardinality of $\Fields^{\rm big}_{F,n,S}$ when $n \geq 4$. For $S$ all of $\cS(F)$, a principle in number theory is that the group $S_n$ is very common, in many rigorous senses. One might at first expect that the sets $\Fields_{F,n,S}$ would behave like smaller versions of the set $\Fields_{F,n,\cS(F)}$, so that most fields in $\Fields_{F,n,S}$ would be in $\Fields^{\rm sym}_{F,n,S}$. This section argues that the evidence points in the opposite direction, at least when one fixes $S$ and considers all $n$ simultaneously. \paragraph*{Ease of constructing fields in $\mbox{\rm $\Fields$}_{F,S}^{\rm small}$.} One has $\Fields_{\Q,S} = \{\Q\}$ for $S = \{\}$ or $S = \{\infty\}$. Otherwise, $\Fields_{\Q,S}$ is infinite, as it at least contains the real cyclotomic fields $\Q[x]/\Phi^+_{p^k}(x)$ for all $p$ in $S$ and all positive $k$. Since the Galois group of $\Q[x]/\Phi^+_{p^k}(x)$ is abelian, these fields are in $\Fields^{\rm small}_{\Q,S}$ whenever their degree is $\geq 4$. There is an elaborate theory for describing the part of $\Fields_{F,S}^{\rm small}$ consisting of fields $K$ such that $\Gal(K^g/\Q)$ is solvable \cite{Ko, NSW}. This theory says that as soon as $S$ is large enough, $\Fields_{F,S}^{\rm small}$ is very large indeed. For example, let $L$ be a maximal pro-2-extension of $\Q$ ramified only within $S = \{\infty,2\}$. Then Markshaitis' theorem \cite[Example 11.18]{Ko} says $\Gal(L/\Q)$ is the free pro-2 product of $\Z/2$ and $\Z_2$. Accordingly $\phi^{\rm small}_{\Q,2^k,\{\infty,2\}}$ grows exponentially with $n = 2^k$. There are also general techniques for constructing non-solvable fields in $\Fields_{F,S}^{\rm small}$. For example using modular forms gives fields with Galois groups with $PSL_2(\F_{\ell^f})$ as a simple subquotient. Already this technique shows that the Galois group corresponding to any $\Fields_{\Q,\{\infty,p,\ell\}}$ has infinitely many simple subquotients different from $A_n$. The $ABC$-construction of \cite{Ro} shows that one can likewise expect infinitely many simple subquotients of the form $PSp_{2k}(\F_\ell)$ involved in $\Fields_{\Q,S}$ for $S$ large enough, e.g.\ $S = \{\infty,2,3\}$. \paragraph*{Difficulty of constructing fields in $\mbox{\rm $\Fields$}_{F,S}^{\rm big}$.} All known constructional techniques for fields with Galois group all of $A_n$ or $S_n$ have only modest control over ramifying primes. The most well-known technique, and one of the best, is uses trinomials. For example, take $F = \Q$ and for $t \in \Q-\{0,1\}$ consider the polynomial \begin{equation} g_{n,t}(x) = x^n - n t x + (n-1) t. \end{equation} Its discriminant is \begin{equation} D_{n,t} = (-1)^{(n-1)(n-2)/2} n^n (n-1)^{n-1} t^{n-1} (t-1). \end{equation} Its Galois group is generically $S_n$ or $A_n$ according to whether or not $D_{n,t}$ is a square. If one chooses $t$ such that the denominator of $t$ and the numerator of $t$ and $t-1$ are only divisible by primes dividing $n(n-1)$, then only these primes can ramify in $K_{n,t} = \Q[x]/g_{n,t}(x)$. The problem here is that only finitely many $n$ keep ramification within any given $S$. Even in the more general context of arbitrary trinomials described in \cite[Section~10]{Ro}, with two relatively prime parameters $n>m$, there are only finitely many $(n,m)$ such that all primes dividing $n m (n-m)$ are within $S$. The recent technique of Chebyshev covers \cite{Ro2} gives larger degree fields, but still to get fields ramified within a given $S$, one needs an appropriate ``numerical accident'' such as $2^3+1=3^2$ for $S = \{\infty,2,3\}$. \paragraph*{The finiteness conjecture.} Based on the considerations just presented, we make the following conjecture. \begin{conjecture} \label{mainconj} Let $F$ be a number field and let $S$ be a finite set of places of $F$. Then the set $\mbox{{\rm $\Fields$}}^{\rm big}_{F,S}$ is finite. \end{conjecture} \noindent In other words, while $\phi_{F,S}^{\rm small}$ is usually infinite, we expect $\phi_{F,S}^{\rm big}$ to always be finite. \paragraph*{Heuristic support for the finiteness conjecture.} Bhargava \cite{Bh} has a heuristic formula for the ``expected number'' of $A_n$ and $S_n$ fields in a given degree $n$ with a given discriminant $d$. The asymptotic behavior of Bhargava's heuristic as $|d| \rightarrow \infty$ agrees with the previously known Davenport-Heilbronn theorem in $n=3$. In fundamental work, Bhargava has proven the analogous theorem for $n=4$ \cite{Bh4} and has announced it for $n=5$, giving one moderate confidence in the heuristic formula for general $n$. Applying Bhargava's heuristic to our situation gives the following ``expected number'' \begin{equation} \label{Bhargava} \phi^{\rm big}_{F,n,S} \approx \frac{1}{2} \prod_{v \in S} \lambda_{F_v,n}, \end{equation} where here we require that all Archimedean places are in $S$. Both $\lambda_{\R,n}$ and $\lambda_{\C,n}$ decrease superexponentially with $n$, according to \eqref{ArchAsymp2}. For each ultrametric place $v$ of $F$, the sequence $\lambda_{F_v,n}$ increases only exponentially, with growth factor $Q_v^{1/(p_v-1)}$. All together, $\prod_{v \in S} \lambda_{F_v,n}$ decreases superexponentially with $n$. This is much more than the mere convergence of $\sum_n \prod_{v \in S} \lambda_{F_v,n}$ which would be enough to heuristically support Conjecture~\ref{mainconj}. Note that one has a heuristic product formula \eqref{Bhargava} only when one appropriately separates by Galois groups. For example, Bhargava's results show that $S_4$ and $D_4$ need to be treated separately. % Bhargava's heuristic requires separating by Galois groups. % In the quartic case, note that $S_4$ and $D_4$ both %have five conjugacy classes. The product heuristic \eqref{Bhargava} %is appropriate if one considers $S_4$ and $A_4$ fields together as %we are doing. One has another product heuristic for %$D_4$, $C_4$, and $V$ together. In our setting, suppose all archimedean %places and also all places of residual characteristic $2$ and $3$ are in $S$. %Then one expects %\begin{align*} %\phi^{\rm big}_{F,4,S} & \approx c_F^{\rm big} 5^{|S|}, & %\phi^{\rm small}_{F,4,S} & \approx c_F^{\rm small} 5^{|S|}, %\end{align*} %with e.g.\, $c_\Q^{\rm big} = 33/100$ and $c_\Q^{\rm small} = 93/200$. %Bhargava's setting of sorting by absolute discriminant is We understand the factor $1/2$ in \eqref{Bhargava} in two different ways, depending on whether $n=2$ or $n \geq 3$. The case $n=2$ is best first explained in the simplified setting $F = \Q$ and $\{\infty,2\} \subseteq S$. Then $\lambda_{\Q_v,2}$ is $1$, $4$, or $2$ according to whether $v$ is $\infty$, $2$, or otherwise. The set $\Fields_{\Q,2,S}$ has exactly $2^{|S|}-1$ elements, each with mass $1/2$. So its total mass is $2^{|S|-1}-2^{-1}$ while \eqref{Bhargava} is $2^{|S|-1}$. The agreement would be perfect if we worked instead with $\Algebras_{\Q,2,S}$ to account for the trivial alternating group $A_2$. For general $F$, the $1/2$ in the case $n=2$ likewise comes from the fact that fields in $\Fields_{F,2,S}$ have mass $1/2$ rather than the usual $1$. For the cases $n \geq 3$, one uses the local signs $(2,d_v) HW(K_v)$ associated to $K/F \in \Algebras_{F,n,S}$ and a place $v \in S$. While these signs are all $1$ in the case $n=2$, in general they can be $1$ or $-1$. For $n \geq 3$, the $1/2$ in the mass formula corresponds to the fact that the product of the local signs is $1$ for any $K$ in $\Fields_{F,n,S}$. For more explicit information on these signs in the case $F = \Q$, see \cite[Section~3.3]{JR}. \paragraph*{Comparison with computational results over $\Q$.} Figure~\ref{compare} summarizes known facts about the case $(F,S) = (\Q,\{\infty,2,3\})$. For $n = 1$, 2, 3, 4, 5, 6, and 7, Jones and Roberts \cite{JR6}, \cite{JR7} evaluated $\phi^{\rm big}_{\Q,\{2,3,\infty\}}$ to $1$, $3.5$, $8.\overline{3}$, $22$, $5$, $54$, and $10$. Roberts \cite{Ro} found more fields in degrees $n = 8, 9$ and also the $S_{32}$ field $\Q[x]/(x^{32} + 2^{16} 3^{5} x^5 + 2^{13} 3^9)$ with discriminant $2^{191} 3^{112}$. Jones is finding more fields in degrees $8$ and $9$ by an ongoing computer search. Malle and Roberts \cite{MR} found 300 more fields in degree $9 \leq n \leq 33$ and discussed the issue of finiteness of $\phi^{\rm big}_{\Q,S}$ noncommittally as an open question. Roberts \cite{Ro2} found $43$ more fields in degrees $12 \leq n \leq 64$. \begin{figure}[htb] \begin{center} \epsfig{file=compare.eps,width=6in} %\includegraphics[width=6in]{compare} \parbox{5in}{\caption{\label{compare} Evaluations (black) and lower bounds (gray) for $\phi^{\rm big}_{\Q,n,\{\infty,2,3\}}$, compared with $\frac{1}{2} \lambda_{\R,n} \lambda_{2,n} \lambda_{3,n}$, with logarithmic vertical scale. }} \end{center} \end{figure} \paragraph*{Low discriminant phenomena and exceptional fields.} Figure~\ref{compare} also compares the above computational results with the more theoretical quantity $\frac{1}{2} \lambda_{\R,n} \lambda_{2,n} \lambda_{3,n}$. Although we are confident in Conjecture~\ref{mainconj} on a qualitative level, the situation remains enigmatic on a quantitative level. We interpret the poor agreement in degrees $\leq 7$ as reflecting the fact that Bhargava's heuristic does not take into account low discriminant phenomena. Experimentally, these low discriminant phenomena always seem to give fewer fields, with the case of cubics quantitatively explained by Roberts \cite{Cubics} using a negative secondary term. Our guess is that the very poor agreement in medium degrees is due to two factors, the same low discriminant phenomena and the incompleteness of the current list of fields. On the other hand, the poor agreement in degree $64$ is in the other direction. We interpret this disagreement as an indication that the constructional method of \cite{Ro2} is very special in nature. Define a field in $\Fields^{\rm big}_{F,n,S}$ to be {\em exceptional} if $\phi^{\rm big}_{F,n',S}<1$ for all $n' \geq n$. The starting point $N(F,S)$ of the exceptional range is not as artificial as may first seem, because the decay of $\phi^{\rm big}_{F,n,S}$ is rapid once it begins. For $F = \Q$ and $S = \{\infty,2,3\}$, $\{\infty,3,5\}$, and $\{\infty,2,5\}$ the exceptional range starts at $N(\Q,S) = 62$, $38$, and $49$ respectively. The field constructed in \cite{Ro2} of degree $100$, Galois group $A_{100}$, and discriminant of the form $3^a 5^b$ is well into the exceptional range. Similarly, the five fields constructed there of degrees $2666$ through $15875$ and discriminant of the form $\pm 2^a 5^b$ are exceptional if, as strongly expected, their Galois groups are the full symmetric group on the degree. \paragraph*{Comparison with computational results over quadratic fields.} In general, let $S$ be a set of rational places containing $\infty$. Let $F$ be any degree $n_0$ number field. For $v$ a place of $\Q$, let $\lambda_{F_v,n} = \prod \lambda_{F_w,n}$, the product being over places $w$ mapping to $v$. Then $\lambda_{F_\infty,n}^{1/n} \sim (e/n)^{n_0}$, independently of the splitting behavior of $\infty$ in $F$. Similarly, $\lambda_{F_p,n}^{1/n} \sim p^{n_0 /(p-1)}$ independently of the splitting behavior of $p$. However, looking at the subexponential factors suggests that $\lambda_{F_v,n}$ is at its lowest if $F_v$ is a field and increases substantially as $F_v$ tends towards the split algebra $\Q_v^{n_0}$. To illustrate this in practice, let $F$ be the field $F_d = \Q(\sqrt{d})$, where $d$ varies over the set $\{-6,-3,-2,-1,2,3,6\}$. Let $S_d \subset \cS(F_d)$ be the set of places mapping to $\infty$, $2$, or $3$ in $\cS(\Q)$. So $|S_d| = 3$ for $d \in \{-6,-3,-1\}$ as none of $\infty$, $2$, or $3$ is split in $F_d$. In contrast, $|S_d|=4$ if $d \in \{-2,2,3,6\}$ as $3$ is split in $F_{-2}$ and $\infty$ is split in the remaining cases. \begin{table}[htb] \begin{center} \parbox{5in}{\caption{\label{relativetable} Evaluations and lower bounds for $\phi^{\rm big}_{\Q(\sqrt{d}),n,\{\infty,2,3\}_d}$ in italics under $d$ compared with $\frac{1}{2} \lambda_{\Q(\sqrt{d})_\infty,n} \lambda_{\Q(\sqrt{d})_2,n} \lambda_{\Q(\sqrt{d})_3,n}$ in plain type under $\prod$.} } \end{center} {\renewcommand{\arraycolsep}{3pt} \[ \begin{array}{|r|rr|rr|rr|rrrr|rr|rrrr|} \cline{8-17} \multicolumn{7}{c|}{\;} & \multicolumn{4}{c|}{\mbox{No $v$ split}} & \multicolumn{2}{c|}{\mbox{$3$ split}} & \multicolumn{4}{c|}{\mbox{$\infty$ split}} \\ \hline n & \C & \R \cdot \R & 4 & 2 \cdot 2 & 9 & 3 \cdot 3 & -6 & -3 & -1 & \prod & -2 & \prod & 2 & 3 & 6 & \prod \\ \hline 2 & 0.500 & 1.000 & 8 & 16 & 2 & 4 & \mathit{3.5} & \mathit{3.5} & \mathit{3.5} &4 & \mathit{7.5} &8& \mathit{7.5} & \mathit{7.5} & \mathit{7.5} & 8 \\ 3 & 0.167 & 0.444 & 9 & 25 & 27 & 81 & \mathit{14.\overline{3}} & \mathit{14.\overline{3}} & \mathit{14.\overline{3}} & 20 & \mathit{46.\overline{3}} & 61 & \mathit{40.\overline{3}} & \mathit{40.\overline{3}} & \mathit{40.\overline{3}} & 54\\ 4 & 0.042 & 0.174 & 272 &1296 & 29 & 121 & \mathit{87} & \mathit{87} &\mathit{87} &164 & \mathit{343} &686 & \mathit{385} & \mathit{385} & \mathit{385} & 685 \\ 5 & 0.008 & 0.047 & 280 & 1600 & 55 & 361 & & \mathit{17} & \mathit{21} & 64 &&421& \mathit{\geq 87} & & & 361 \\ \hline \end{array} \] } \end{table} Table~\ref{relativetable} compares the two sides of \eqref{Bhargava} in degrees $2 \leq n \leq 5$. The first pair of columns gives $\lambda_{\C,n} < \lambda_{\R,n}^2$ to three decimal places. The next two pairs of columns likewise give $\lambda_{4,n} < \lambda_{2,n}^2$ and $\lambda_{9,n} < \lambda_{3,n}^2$. The column $\lambda_{2,n}^2$ is given for the sake of uniformity, but is not needed as $2$ does not split in any of the $F_d$. Next follow masses $\phi^{\rm big}_{\Q(\sqrt{d}),n,\{\infty,2,3\}_d}$ in italics, with the corresponding product $\frac{1}{2} \lambda_{\Q(\sqrt{d})_\infty,n} \lambda_{\Q(\sqrt{d})_2,n} \lambda_{\Q(\sqrt{d})_3,n}$ rounded to the nearest integer in regular type to its right. The italicized masses are computed from \cite{JR6} for $n=2,3$ fields and from \cite{Dr} and associated ongoing searches for $n=4,5$. For each $n$, the references list degree $2n$ extensions of $\Q$ with a corresponding degree $2n$ permutation group $G$. We extracted those with subfields isomorphic to $F_d$. Each such field $K$ gives either a single extension of $F_d$ or two conjugate extensions, according to whether the Galois closure $K^g$ of $K$ over $F_d$ is Galois or not over $\Q$. The Galois groups $G$ giving one and two fields are as follows: \begin{equation*} \begin{array}{r | l | l} 2n & \mbox{One field} & \mbox{Two conjugate fields} \\ \hline 4 & C_4, V & D_4 \\ 6 & A_3C_2^*, \; S_3^tC_2^*, \; S_3C_2 & T5^*, \; T9,\; T10,\; T13 \\ 8 & A_4C_2^*, \; S_4^tC_2^*, \; S_4C_2& T33^*,\; T34,\; T41, \; T42^*,\; T45, \; T46, \; T47\\ 10 & A_5C_2^*, \; S_5^tC_2^*, \; S_5C_2& T40^*,\; T41,\; T42,\; T43 \\ \end{array} \end{equation*} Here the last group on each list is always $S_n^2.C_2$. The starred groups are the ones giving rise to $\Gal(K^g/\Q(\sqrt{d})) \cong A_n$. This distinction plays a role in the construction of Table~\ref{relativetable} only for $n=3$, as in this case $A_3$ fields are counted with mass $1/3$; if we were counting the total number of fields then the $n=3$ entries left to right would be $\mathit{17}$, $\mathit{23}$, $\mathit{17}$, $\mathit{49}$, $\mathit{41}$, $\mathit{41}$, and $\mathit{43}$. Table~\ref{relativetable} reflects again how \eqref{Bhargava} does not take into account low discriminant phenomena and consequently $\frac{1}{2} \prod_{v \in S} \lambda_{\Q(\sqrt{d})_v,n}$ is an overestimate for the total mass $\phi^{\rm big}_{F_d,n,S_d}$. However, Table~\ref{relativetable} also illustrates a uniformity in the overestimation, as in degrees $2$, $3$, $4$, and $5$ the factor is approximately $1$, $1.3$, $2$, and $3.5$, independent of $d$. Thus the principle that splitting primes in the base field increase the number of fields is clearly visible at the level of actual fields. \section{Contrast with positive characteristic} \label{Contrast} It is standard in number theory to talk about global fields, meaning either number fields as in Section~\ref{NumberFields} or function fields in one variable over a finite field. By the latter, one means fields $F$ which can be presented as finite extensions of some rational function field $\F_q(t)$. One can talk uniformly about completions of these global fields, getting local fields. In characteristic zero, the local fields are exactly the ones considered in Sections~\ref{RC}-\ref{adic2}. In positive characteristic $p$, all local fields are isomorphic to the Laurent series field $\F_{q}((u))$, for $q$ some power of $p$. Often in number theory, the characteristic zero and the positive characteristic situations are quite similar; this principle is well illustrated by our comments on the Serre mass formula below. However with regards to both our specialization $(y,z) = (1,1)$ and our finiteness conjecture, the case of positive characteristic presents a sharp contrast. \paragraph*{Allowing $e_0=\infty$.} To accommodate positive characteristic, Sections~\ref{Wild} and \ref{Three} can be extended by allowing $e_0 = \infty$ also. Ore numbers in this situation are simpler. Namely, as before, $\Ore(p,\infty,e)$ is $\{0\}$ in the tame case when $e$ is not a multiple of $p$. However in the complementary wild case, the Ore table consists of a single block with infinitely many rows. Thus $\Ore(p,\infty,e)$ is always the complete set of positive integers which are not multiples of $p$; this is a radical difference because now the number of Ore numbers is infinite. The dimension associated to an Ore number $s$ is simply $d(p,\infty,e,s) = \lceil s/p \rceil$. Since all Ore numbers $s$ are non-maximal, $W(p,\infty,e,s)$ always consists of the vectors in $\overline{\F}_p^{d(p,\infty,e,s)}$ with first component nonzero. The function $\Lambda_{p,\infty,f_0}(x,y,z)$ is defined as before. Writing $s = s_0 p + s_1$, with $1 \leq s_1 \leq p-1$, it takes the explicit form \begin{equation} \label{charpLambda} \Lambda_{p,\infty,f_0}(x,y,z) = \prod_{s_0=0}^\infty \prod_{s_1 = 1}^{p-1} \frac{1 - q^{s_0} x^e y^{e-1} z^{p s_0 + s_1}}{1 - q^{s_0+1} x^e y^{e-1} z^{p s_0 + s_1}}, \end{equation} with $q = p^{f_0}$, as always. \paragraph*{A contrast between the two specializations.} The two situations in Section~\ref{One} now present a sharp contrast. The Serre mass formula continues to hold: $\Lambda_{p,\infty,f_0}(x,1,1/q) = \Lambda(x)$, with essentially the same proof. Our formula $\Lambda_{p,\infty,f_0}(x,1,1) = \Lambda_Q(x)$ becomes completely degenerate as $Q = p^{e_0f_0}$ needs to be regarded as $\infty$. The coefficient of $x^n$ on both sides is $\lambda_{n}$ if $n1$ we expect that ``almost all'' specializations of the family \eqref{genform} to extensions of $\F_p(t)$ have Galois group all of $A_n$ or $S_n$. Another simple specialization of \eqref{genform} is $g_{p,k}(x) = x^{k p} - x + t$ for $p$ a prime and $k$ a positive integer. Changing variables via $x = 1/y$ and $t = 1/s$, the equation becomes $y^{k p} - s y^{k p -1} + s$. This is an Eisenstein polynomial with conductor $c = c_t + c_w = (kp-1) + (kp-1) = 2 k p - 2$. Thus $\F_p(t)[x]/g_{p,k}(x)$ is the function field of a genus zero curve, a fact which is also clear by the single global observation that $F[x]/g_{p,k}(x) = \F_p(x)$, as $t = - x^{k p} - x$. If $k = p^j$ then one has $g_{p,p^j}(x) = h_j(x) + t$ with $h_j(x)$ the Artin-Schreier polynomial $h_1(x) = x^p-x$ composed with itself $j$ times. Thus the Galois group of $g_{p,p^j}(x)$ is solvable. In the complementary case, computations with Frobenius elements using Jordan's criterion \eqref{Jordan} suggest that the Galois group of $g_{p,k}(x)$ is all of $A_{pk}$ or $S_{pk}$ except in the cases $g_{3,4}(x) = x^{12} - x + t$ and $g_{2,12}(x) = x^{24} - x + t$. Here the Galois groups are known to be the Mathieu group $M_{11}$ in its degree 12 representation and the Mathieu group $M_{24}$, respectively \cite[Theorems~6.6 and 6.3]{Ab}. So the evidence is strong that for each $p$, the field $\F_p(t)[x]/g_{p,k}(x)$ is in $\Fields^{\rm big}_{\F_p(t),pk,\{\infty\}}$ for infinitely many $k$. Abhyankar has studied many similar genus zero families; typically the focus is extracting rare examples with small Galois groups from families with generic Galois group $A_n$ or $S_n$. The same heuristic which supports Conjecture~\ref{mainconj} in characteristic zero gives two reasons why the corresponding statement fails in positive characteristic. Again one can interpret $\frac{1}{2} \prod_{v \in S} \lambda_{F_v,n}$ as the expected total mass of fields in $\Fields^{\rm big}_{F,n,S}$. But in the function field case if $n \geq p$ then each of the factors $\lambda_{F_v,n}$ is itself infinite; this is the phenomenon behind the existence of the family $g_{k}(x)$ above. Even if one bounds ramification somehow so that each $\lambda_{F_v,n}$ is replaced by a finite number $\lambda^*_{F_v,n}$, the numbers $\frac{1}{2} \prod_{v \in S} \lambda^*_{F_v,n}$ can still increase due to the lack of Archimedean places. This is the phenomenon behind the existence of the family of polynomials $g_{p,k}(x)$. \begin{thebibliography}{4} \bibitem{Ab} S. Abhyankar, Mathieu group coverings and linear group coverings, in M. D. 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Sloane, {\em The On-Line Encyclopedia of Integer Sequences} at \\ \href{http://www.research.att.com/~njas/sequences/}{http://www.research.att.com/$\sim$njas/sequences/} \bibitem{St} R.\ P.\ Stanley, {\em Enumerative Combinatorics}, vol. 2, Cambridge, 1999. \end{thebibliography} \bigskip \hrule \bigskip \noindent 2000 {\it Mathematics Subject Classification}: Primary 11S15; Secondary 11P72, 11R21.\\ \noindent {\it Keywords}: wild, partition, $p$-adic, ramified, mass. \bigskip \hrule \bigskip \noindent (Concerned with sequences \seqnum{A000041}, \seqnum{A000085}, \seqnum{A010054},\seqnum{A033687}, \seqnum{A131139}, and \seqnum{A131140}.) \bigskip \hrule \bigskip \vspace*{+.1in} \noindent Received March 28 2007; revised version received June 18 2007. Published in {\it Journal of Integer Sequences}, June 18 2007. \bigskip \hrule \bigskip \noindent Return to \htmladdnormallink{Journal of Integer Sequences home page}{http://www.cs.uwaterloo.ca/journals/JIS/}. \vskip .1in \end{document}